As President Adams was about to leave office he began a flurry of appointing Federalist judges (judges were more openly political then) up until the night of March 3, 1801. Jefferson, a Democratic Republican took office on March 4th.
One of Adam's "midnight judges" as they became known was Federalist William Marbury
President Jefferson had his secretary of state James Madison dismiss William Marbury from his post as a judge in Washington, D.C.
Marbury sued to get his job back. The lawsuit reached the Supreme Court in 1804, known as Marbury V. Madison
Justice John Marshall (Federalist) who led the Supreme Court, dismissed the case saying that the case was based on the Judiciary Act of 1789 which, according to Marshall and the Supreme Court, was unconstitutional.
Jefferson and the Democratic Republicans were happy that a Federalist judge (Marshall) ruled against another Federalist (Marbury)
In the short term this makes the Democratic Republicans happy, but in the long term this sets the precedent that the Supreme Court can challenge something Congress has done (Judiciary Act 1789) as unconstitutional. Therefore the Federalist Supreme court could "flex its muscle" and rule anything the Democratic-Republican Congress did as unconstitutional.
"Marshall lost the battle, but wins the political war" over the Dem. Republicans
Supreme Court challenges to Congress are known as "judicial review"
The Judiciary Act of 1789
What the Judiciary Act of 1789 means